Quantum isometry group of dual of finitely generated discrete groups - II

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BLAISE PASCAL

Arnab Mandal

Quantum isometry group of dual of finitely generated discrete groups - II

Volume 23, n^o2 (2016), p. 219-247.

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Quantum isometry group of dual of finitely generated discrete groups - II

Arnab Mandal

Abstract

As a continuation of the programme of [13], we carry out explicit computations ofQ(Γ, S), the quantum isometry group of the canonical spectral triple onCr^∗(Γ) coming from the word length function corresponding to a finite generating set S, for several interesting examples of Γ not covered by the previous work [13]. These include the braid group of 3 generators,Z^∗n4 etc. Moreover, we give an alternative description of the quantum groupsH_s⁺(n,0) andK_n⁺ (studied in [3], [4]) in terms of free wreath product. In the last section we give several new examples of groups for whichQ(Γ) turns out to be a doubling ofC^∗(Γ).

1. Introduction

It is a very important and interesting problem in the theory of quantum groups and noncommutative geometry to study ‘quantum symmetries’ of various classical and quantum structures. S.Wang pioneered this by defining quantum permutation groups of finite sets and quantum automorphism groups of finite dimensional matrix algebras. Later on, a number of mathematicians including Wang, Banica, Bichon and others ([1], [8], [20]) developed a theory of quantum automorphism groups of finite dimensional C^∗-algebras as well as quantum isometry groups of finite metric spaces and finite graphs. In [11] Goswami extended such constructions to the set-up of possibly infinite dimensional C^∗-algebras, and more interestingly, that of spectral triples a la Connes [10], by defining and studying quantum isometry groups of spectral triples. This led to the study of such quantum isometry groups by many authors including Goswami, Bhowmick, Skalski, Banica, Bichon, Soltan, Das, Joardar and others. In the present paper, we are focusing on a particular class of spectral triples, namely those coming from the word-length metric of finitely generated discrete

Keywords:Compact quantum group, Quantum isometry group, Spectral triple.

Math. classification:58B34, 46L87, 46L89.

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groups with respect to some given symmetric generating set. There have been several articles already on computations and study of the quantum isometry groups of such spectral triples, e.g. [3], [4], [7], [14], [18] and refer- ences therein. In [13] together with Goswami we also studied the quantum isometry groups of such spectral triples in a systematic and unified way.

Here we compute Q(Γ, S) for more examples of groups including braid groups, Z4∗Z4· · · ∗Z4

| {z }

n copies

etc.

The paper is organized as follows. In Section 2 we recall some defini- tions and facts related to compact quantum groups, free wreath product by quantum permutation group and quantum isometry group of spectral triples defined by Bhowmick and Goswami in [6]. This section also contains the doubling procedure of a compact quantum group, say Q, with respect to an order 2 CQG automorphism θ. The doubling is denoted by D_θ(Q). In Section 3 we compute Q(Γ, S) for braid group with 3 generators. Its underlying C^∗-algebra turns out to be four direct copies of the group C^∗-algebra. In fact, it is precisely a doubling of doubling of the group C^∗-algebra. Section 4 contains an interesting description of the quantum groups H_s⁺(n,0) and K_n⁺ (studied in [3], [4]) in terms of free wreath product. Moreover, Q(Γ, S) is computed for Γ =Z4∗Z4· · · ∗Z4

| {z }

n copies

. In the last section we present more examples of groups as in [14], [18], Section 5 of [13] where Q(Γ, S) turns out to be a doubling ofC^∗(Γ).

2. Preliminaries

First of all, we fix some notational conventions which will be useful for the rest of the paper. Throughout the paper, the algebraic tensor product and the spatial (minimal) C^∗-tensor product will be denoted by ⊗ and

⊗ˆ respectively. We’ll use the leg-numbering notation. Let Q be a unital C^∗-algebra. Consider the multiplier algebra M(K(H) ˆ⊗Q) which has two natural embeddings into M(K(H) ˆ⊗Q⊗Q). The first one is obtained byˆ extending the mapx7→x⊗1 and the second one is obtained by composing this map with the flip on the last two factors. We will write ω¹² and ω¹³ for the images of an element ω ∈ M(K(H) ˆ⊗Q) under these two maps respectively. We’ll denote the HilbertC^∗-module byH⊗Q¯ obtained by the

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completion of H ⊗ Q with respect to the norm induced by the Q valued inner product ξ⊗q, ξ⁰⊗q⁰ :=<ξ, ξ⁰>q^∗q⁰, whereξ, ξ⁰ ∈ H, q, q⁰ ∈ Q.

2.1. Compact quantum groups and free wreath product Let us recall the basic notions of compact quantum groups, then actions on C^∗-algebra and free wreath product by quantum permutation groups.

Definition 2.1. A compact quantum group (CQG for short) is a pair (Q,∆), where Q is a unital C^∗-algebra and ∆ : Q → Q⊗Qˆ is a unital C^∗-homomorphism satisfying two conditions:

(1) (∆⊗id)∆ = (id⊗∆)∆ (co-associativity ).

(2) Each of the linear spans of ∆(Q)(1⊗ Q) and that of ∆(Q)(Q ⊗1) is norm dense in Q⊗Q.ˆ

A CQG morphism from (Q₁,∆₁) to another (Q₂,∆₂) is a unital C^∗- homomorphism π :Q₁ 7→ Q₂ such that (π⊗π)∆₁ = ∆₂π.

Definition 2.2. (Q₁,∆₁) is called a quantum subgroup of (Q₂,∆₂) if there exists a surjective C^∗-morphism η from Q₂ to Q₁ such that (η ⊗ η)∆₂ = ∆₁η holds.

Sometimes we may denote the CQG (Q,∆) simply asQ, if ∆ is understood from the context.

Definition 2.3. A unitary (co) representation of a CQG (Q,∆) on a Hilbert space H is a C-linear map from H to the Hilbert module H⊗Q¯ such that

(1) U(ξ), U(η)=<ξ, η>1Q whereξ, η ∈ H.

(2) (U ⊗id)U = (id⊗∆)U.

(3) Span{U(ξ)b:ξ ∈ H, b∈ Q} is dense inH⊗Q.¯

Given such a unitary representation we have a unitary element ˜U be- longing to M(K(H) ˆ⊗Q) given by ˜U(ξ ⊗b) = U(ξ)b,(ξ ∈ H, b ∈ Q) satisfying (id⊗∆)( ˜U) = ˜U¹²U˜¹³.

Here we state Proposition 6.2 of [15] which will be useful for us.

Proposition 2.4. If a unitary representation of a CQG leaves a finite dimensional subspace of H, then it’ll also leave its orthogonal complement invariant.

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Remark 2.5. It is known from [21] that the linear span of matrix elements of a finite dimensional unitary representation form a dense Hopf *-algebra Q₀ of (Q,∆), on which an antipodeκ and co-unitare defined.

Definition 2.6. We say that a CQG (Q,∆) acts on a unital C^∗-algebra B if there is a unital C^∗-homomorphism (called action) α : B → B⊗Qˆ satisfying the following :

(1) (α⊗id)α= (id⊗∆)α.

(2) Linear span ofα(B)(1⊗ Q) is norm dense in B⊗Q.ˆ

Definition 2.7. The action is said to be faithful if the∗-algebra generated by the set {(f ⊗id)α(b) ∀ f ∈B^∗, ∀ b ∈B} is norm dense in Q, where B^∗ is the Banach space dual of B.

Remark 2.8. Given an action α of a CQG Q on a unital C^∗-algebra B, we can always find a norm-dense, unital ∗-subalgebra B₀ ⊆ B such that α|_B₀ :B0 7→B0⊗ Q₀ is a Hopf-algebraic co-action. Moreover,α is faithful if and only if the∗-algebra generated by{(f⊗id)α(b)∀f ∈B₀^∗, ∀b∈B₀} is the whole of Q₀.

Given two CQG’s Q₁,Q₂ the free productQ₁?Q₂ admits the natural CQG structure equipped with the following universal property (for more details see [19]):

Proposition 2.9.

(i) The canonical injections, say i1, i2, from Q₁ and Q₂ to Q₁?Q₂ are CQG morphisms.

(ii) Given any CQG C and morphisms π1 :Q₁ 7→ C and π2 :Q₂ 7→ C there always exists a unique morphism denoted by π := π₁ ∗π₂ from Q₁?Q₂ to C satisfying π◦ik=πk for k= 1,2.

Definition 2.10. The C^∗-algebra underlying the quantum permutation group, denoted by C(S_N⁺) is the universal C^∗-algebra generated by N² elements tij such that the matrix ((tij)) is unitary with

t_ij =t^∗_ij =t²_ij ∀i, j , X

i

t_ij = 1 ∀j, ^X

j

t_ij = 1 ∀i , tijt_ik = 0, tjit_ki= 0 ∀ i, j, k with j6=k .

It has a coproduct ∆ is given by ∆(tij) = Σ^N_k=1t_ik ⊗t_kj, such that (C(S_N⁺),∆) becomes a CQG.

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For further details see [20]. We also recall from [9] the following:

Definition 2.11. Let Q be a compact quantum group and N >1. The free wreath product of Q by the quantum permutation group C(S_N⁺), is the quotient of Q^?N ? C(S_N⁺) by the two sided ideal generated by the elements

ν_k(a)t_ki−t_kiν_k(a), 1≤i, k≤N, a∈ Q,

where ((tij)) is the matrix coefficients of the quantum permutation group C(S_N⁺) andν_k(a) denotes the natural image ofa∈ Qin the k-th factor of Q^?N. This is denoted by Q ∗_wC(S_N⁺).

Furthermore, it admits a CQG structure, where the comultiplication satisfies

∆(νi(a)) =

N

X

k=1

νi(a₍₁₎)tik⊗νk(a₍₂₎).

Here we have used the Sweedler convention of writing ∆(a) =a₍₁₎⊗a₍₂₎. 2.2. Some facts about quantum isometry groups

First of all, we are defining the quantum isometry group of spectral triples defined by Bhowmick and Goswami in [6].

Definition 2.12. Let (A^∞,H,D) be a spectral triple of compact type (a la Connes). Consider the category Q(D) ≡ Q(A^∞,H,D) whose objects are (Q, U) where (Q,∆) is a CQG having a unitary representation U on the Hilbert space Hsatisfying the following:

(1) ˜U commutes with (D ⊗1_Q).

(2) (id⊗φ)◦ad_U_˜(a)∈(A^∞)⁰⁰for alla∈ A^∞and φis any state on Q, wheread_U_˜(x) := ˜U(x⊗1) ˜U^∗ forx∈ B(H).

A morphism between two such objects (Q, U) and (Q⁰, U⁰) is a CQG morphism ψ :Q → Q⁰ such that U⁰ = (id⊗ψ)U. If a universal object exists in Q(D) then we denote it by QISO⁺^(A^∞,H,D) and the corresponding largest Woronowicz subalgebra for which ad_U_˜

0 is faithful, whereU₀ is the unitary representation ofQISO⁺^(A^∞,H,D), is called the quantum group of orientation preserving isometries and denoted by QISO⁺(A^∞,H,D).

Let us state Theorem 2.23 of [6] which gives a sufficient condition for the existence of QISO⁺(A^∞,H,D).

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Theorem 2.13. Let (A^∞,H,D) be a spectral triple of compact type. As- sume that D has one dimensional kernel spanned by a vectorξ ∈ Hwhich is cyclic and separating forA^∞ and each eigenvector ofDbelongs toA^∞ξ.

Then QISO⁺(A^∞,H,D) exists.

Let (A^∞,H,D) be a spectral triple satisfying the condition of Theo- rem 2.13 and A₀₀ =Lin{a∈ A^∞:aξ is an eigenvector of D}. Moreover, assume that A₀₀ is norm-dense in A^∞. Let ˆD:A₀₀ 7→ A₀₀ be defined by D(a)ξˆ =D(aξ) (a∈ A₀₀). This is well defined asξis cyclic and separating vector for A^∞. Let τ be the vector state corresponding to the vectorξ.

Definition 2.14. LetAbe aC^∗-algebra andA^∞be a dense *-subalgebra such that (A^∞,H,D) is a spectral triple as above. Let ˆC(A^∞,H,D) be the category with objects (Q, α) such that Q is a CQG with aC^∗-action α on A such that

(1) α isτ preserving, i.e. (τ ⊗id)α(a) =τ(a).1 for all a∈ A.

(2) α maps A₀₀ intoA₀₀⊗ Q.

(3) αDˆ = ( ˆD ⊗I)α.

The morphisms in ˆC(A^∞,H,D) are CQG morphisms intertwining the respective actions.

Proposition 2.15. It is shown in Corollary 2.27 of[6]thatQISO⁺(A^∞, H,D) is the universal object in C(Aˆ ^∞,H,D).

2.3. QISO for a spectral triple on C_r^∗(Γ)

Now we discuss the special case of our interest. Let Γ be a finitely generated discrete group with generating set S={a₁, a⁻¹₁ , a2, a⁻¹₂ ,· · ·a_k, a⁻¹_k }.

We make the convention of choosing the generating set to be symmetric, i.e. ai ∈ S implies a⁻¹_i ∈ S ∀ i. In case some ai has order 2, we include only a_i, i.e. not count it twice. The corresponding word length function on the group defined by l(g) = min {r ∈ N, g = h1h2· · ·hr} where hi ∈ S i.e. for each i, hi = aj or a⁻¹_j for some j. Notice that S = {g ∈ Γ, l(g) = 1}, using this length function we can define a metric on Γ by d(a, b) = l(a⁻¹b) ∀ a, b ∈ Γ. This is called the word metric corresponding to the generating set S. Now consider the algebra C_r^∗(Γ), which is the C^∗-completion of the group ring CΓ viewed as a subalgebra of B(l²(Γ)) in the natural way via the left regular representation. We define a Dirac operator DΓ(δg) = l(g)δg. In general, DΓ is an unbounded

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operator.

Dom(DΓ) ={ξ∈l²(Γ) : ^X

g∈Γ

l(g)²|ξ(g)|² <∞}.

Here, δ_g is the vector inl²(Γ) which takes value 1 at the pointg and 0 at all other points. Natural generators of the algebra CΓ (images in the left regular representation ) will be denoted by λ_g, i.e. λ_g(δ_h) = δ_gh. Let us define

Γ_r ={δ_g |l(g) =r}, Γ≤r ={δ_g |l(g)≤r}.

Moreover,prandqrbe the orthogonal projections ontoSp(Γr) andSp(Γ≤r) respectively. Clearly

D_Γ= ^X

n∈N0

np_n,

wherepr=qr−q_r−1andp0 =q0. The canonical trace onC_r^∗(Γ) is given by τ(^Pc_gλ_g) =c_e. It is easy to check that (CΓ,l²(Γ), D_Γ) is a spectral triple.

Now takeA=C_r^∗(Γ), A^∞=CΓ, H=l²(Γ) andD=D_Γ as before. Then QISO⁺(CΓ, l²(Γ), D_Γ) exists by Theorem 2.13, taking δ_e as the cyclic separating vector for CΓ. As the object depends on the generating set of Γ it is denoted by Q(Γ, S). Most of the times we denote it by Q(Γ) if S is understood from the context. As in [7] its action α (say) on C_r^∗(Γ) is determined by

α(λ_γ) = ^X

γ⁰∈S

λ_γ⁰ ⊗q_γ,γ⁰,

where the matrix [q_γ,γ⁰]_γ,γ⁰_∈S is called the fundamental representation in M_card(S)(Q(Γ, S)). Note that we have ∆(q_γ,γ⁰) =^P_βq_β,γ⁰ ⊗qγ,β.

Q(Γ, S) is also the universal object in the category ˆC(CΓ,l²(Γ), D_Γ) by Proposition 2.15 and observe that all the eigenspaces of ˆD_Γ, where ˆD_Γ as in Definition 2.14, are invariant under the action. The eigenspaces of ˆD_Γ are precisely the setSpan{λ_g |l(g) =r}with r≥0.

It can also be identified with the universal object of some other categories naturally arising in the context. Consider the categoryC_τ of CQG’s consisting of the objects (Q, α) such that α is an action of Q on C_r^∗(Γ) satisfying the following two properties:

(1) α leaves Sp(Γ1) invariant.

(2) It preserves the canonical traceτ of C_r^∗(Γ).

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Morphisms inC_τ are CQG morphisms intertwining the respective actions.

Lemma 2.16. The two categories C_τ andC(ˆ CΓ, l²(Γ), D_Γ) are isomor- phic.

Proof. Let (Q, α)∈C(ˆ CΓ, l²(Γ), D_Γ) then clearly (Q, α)∈C_τ. Consider any (Q, α)∈C_τ. Then the actionα leavesSp(Γ≤r) invariant ∀r≥2 as it is an algebra homomorphism and it leavesSp(Γ₁) invariant. Consider the linear mapU(x) :=α(x) fromC_r^∗(Γ)⊂ H=l²(Γ) to H⊗Q¯ is an isometry by the invariance of τ. Thus it extends to H and in fact it becomes a unitary representation. Now, observe that Sp(Γ_r) is the orthogonal complement of Sp(Γ≤r−1) inside Sp(Γ≤r). By the Proposition 2.4, Sp(Γr) is invariant under U too, i.e. α leaves Span{λ_g |l(g) =r} invariant for all r. Thus (Q, α)∈C(ˆ CΓ,l²(Γ), D_Γ). Clearly any morphism in the category C_τ is in the category ˆC(CΓ,l²(Γ), D_Γ) and vice-versa. This completes the

proof.

Corollary 2.17. It follows from Lemma 2.16 that there is a universal object, say (Q_τ, α_τ) in C_τ and (Q_τ, α_τ)∼=Q(Γ, S).

We now identify Q(Γ, S) as a universal object in yet another category.

Let us recall the quantum free unitary group Au(n) introduced in [19].

It is the universal unital C^∗-algebra generated by ((aij)) subject to the conditions that ((a_ij)) and ((a_ji)) are unitaries. Moreover, it admits a co-product structure with comultiplication ∆(aij) = Σⁿ_l=1a_lj ⊗a_il. Con- sider the category C with objects (C,{x_ij, i, j = 1,· · · ,2k}) where C is a unital C^∗-algebra generated by ((xij)) such that ((xij)) as well as ((x_ji)) are unitaries and there is a unital C^∗- homomorphism αC from C_r^∗(Γ) to C_r^∗(Γ) ˆ⊗C sending ei to ^P^2k_j=1ej ⊗xij, where e2i−1 = λai and e_2i =λ⁻¹_a_i ∀ i= 1,· · · , k. The morphisms from (C,{x_ij, i, j = 1,· · ·,2k}) to (P,{p_ij, i, j= 1,· · ·,2k}) are unital∗-homomorphismsβ:C 7→ P such that β(x_ij) =p_ij.

Moreover, by definition of each object (C,{x_ij, i, j = 1,· · · ,2k}) we get a unital ∗-morphism ρC from A_u(2k) to C sendinga_ij tox_ij. Let the kernel of this map be IC and I be intersection of all such ideals. Then CÛ := Au(2k)/I is the universal object generated by xÛ_ij in the category C. Furthermore, we can show, following a line of arguments similar to those in Theorem 4.8 of [12], that it has a CQG structure with the coproduct ∆(xÛ_ij) =^P_lxÛ_lj⊗xÛ_il.

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Proposition 2.18. (Q_τ, ατ) andC^U are isomorphic as CQG.

For the proof of the above proposition, the reader is referred to Propo- sition 2.15 of [13]. Now we fix some notational conventions which will be useful in later sections. Note that the action α is of the form

α(λ_a₁) =λ_a₁⊗A₁₁+λ_a⁻¹

1

⊗A₁₂+λ_a₂ ⊗A₁₃+λ_a⁻¹

2

⊗A₁₄ +· · ·+λ_a_k⊗A_1(2k−1)+λ_a⁻¹

k

⊗A_1(2k), α(λ_a⁻¹

1

) =λ_a₁⊗A^∗₁₂+λ_a⁻¹

1

⊗A^∗₁₁+λ_a₂ ⊗A^∗₁₄+λ_a⁻¹

2

⊗A^∗₁₃ +· · ·+λ_a_k⊗A^∗_1(2k)+λ_a⁻¹

k

⊗A^∗_1(2k−1), α(λa2) =λa1⊗A₂₁+λ_a⁻¹

1

⊗A₂₂+λa2 ⊗A₂₃+λ_a⁻¹

2

⊗A₂₄ +· · ·+λ_a_k⊗A_2(2k−1)+λ_a⁻¹

k

⊗A_2(2k), α(λ_a⁻¹

2

) =λa1⊗A^∗₂₂+λ_a⁻¹

1

⊗A^∗₂₁+λa2 ⊗A^∗₂₄+λ_a⁻¹

2

⊗A^∗₂₃ +· · ·+λ_a_k⊗A^∗_2(2k)+λ_a⁻¹

k

⊗A^∗_2(2k−1) ... ...

α(λ_a_k) =λ_a₁⊗A_k1+λ_a⁻¹

1

⊗A_k2+λ_a₂ ⊗A_k3+λ_a⁻¹

2

⊗A_k4 +· · ·+λak⊗Ak(2k−1)+λ_a⁻¹

k

⊗A_k(2k), α(λ_a⁻¹

k

) =λ_a_k⊗A^∗_k2+λ_a⁻¹

1

⊗A^∗_k1+λ_a₂ ⊗A^∗_k4+λ_a⁻¹

2

⊗A^∗_k3 +· · ·+λa_k⊗A^∗_k(2k)+λ_a⁻¹

k

⊗A^∗_k(2k−1). From this we get the unitary representation

U ≡((u_ij)) =







A11 A12 A13 A14 · · · A1(2k−1) A_1(2k) A^∗₁₂ A^∗₁₁ A^∗₁₄ A^∗₁₃ · · · A^∗_1(2k) A^∗_1(2k−1) A21 A22 A23 A24 · · · A2(2k−1) A_2(2k) A^∗₂₂ A^∗₂₁ A^∗₂₄ A^∗₂₃ · · · A^∗_2(2k) A^∗_2(2k−1)

... ...

A_k1 A_k2 A_k3 A_k4 · · · A_k(2k−1) A_k(2k) A^∗_k2 A^∗_k1 A^∗_k4 A^∗_k3 · · · A^∗_k(2k) A^∗_k(2k−1)





 .

From now on, we call it as fundamental unitary. The coefficients Aij and A^∗_ij’s generate a norm dense subalgebra ofQ(Γ, S). We also note that the antipode of Q(Γ, S) maps uij tou^∗_ji.

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Remark 2.19. Using Corollary 2.17 and Proposition 2.18, Q(Γ, S) is the universal unital C^∗-algebra generated by A_ij as above subject to the re- lations that U is a unitary as well as U^t and α given above is a C^∗- homomorphism on C_r^∗(Γ).

2.4. Q(Γ) as a doubling of certain quantum groups

In this subsection we briefly recall from [14], [17] the doubling procedure of a compact quantum group which is just a particular case of a smash co-product, a well-known construction of Hopf-algebra theory introduced in [16]. Let (Q,∆) be a CQG with a CQG-automorphism θ such that θ² = id. The doubling of this CQG, say (D_θ(Q),∆) is given˜ by D_θ(Q) := Q ⊕ Q (direct sum as a C^∗-algebra), and the coproduct is defined by the following, where we have denoted the injections of Q onto the first and second coordinate in D_θ(Q) byξ andη respectively, i.e.

ξ(a) = (a,0), η(a) = (0, a), (a∈ Q).

∆˜ ◦ξ= (ξ⊗ξ+η⊗[η◦θ])◦∆,

∆˜ ◦η= (ξ⊗η+η⊗[ξ◦θ])◦∆.

It is known from [17] that, if there exists a non trivial automorphism of order 2 which preserves the generating set, then D_θ(C^∗(Γ)) ([14], [17]) will be always a quantum subgroup of Q(Γ). Below we give some sufficient conditions for the quantum isometry group to be a doubling of some CQG. For this, it is convenient to use a slightly different notational convention: let U2i−1,j = A_ij for i = 1, . . . , k, j = 1, . . . ,2k and U_2i,2l=A^∗_i(2l−1), U2i,2l−1 =A^∗_i(2l) fori= 1, . . . , k, l= 1, . . . , k.

Proposition 2.20. Let Γ be a group with k generators {a₁, a₂,· · ·a_k} and define γ_2l−1 :=a_l, γ_2l :=a⁻¹_l ∀ l= 1,2,· · · , k. Now σ be an order 2 automorphism on the set {1,2,· · · ,2k−1,2k} andθbe an automorphism of the group given by θ(γ_i) = γ_σ(i) ∀ i = 1,2,· · ·,2k. We assume the following:

(1) B_i :=U_i,σ(i)6= 0 ∀ i, and U_i,j = 0 ∀ j 6∈ {σ(i), i},

(2) A_iB_j=B_jA_i= 0∀i, jsuch thatσ(i)6=i, σ(j)6=j,whereA_i=U_i,i, (3) All U_i,jU_i,j^∗ are central projections,

(4) There are well defined C^∗-isomorphisms π₁, π₂ from C^∗(Γ) to C^∗{A_i, i = 1,2,· · ·,2k} and C^∗{B_i, i = 1,2,· · ·,2k} respectively

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such that

π1(λai) =Ai, π2(λai) =Bi ∀i.

ThenQ(Γ)is doubling of the group algebra (i.e.Q(Γ)∼=D_θ(C^∗(Γ))) corre- sponding to the given automorphismθ. Moreover, the fundamental unitary takes the following form







A₁ 0 0 0 · · · 0 B₁

0 A2 0 0 · · · B2 0

0 0 A₃ 0 · · · 0 0

0 0 0 A₄ · · · 0 0

... ...

0 B2k−1 0 0 · · · A2k−1 0

B2k 0 0 0 · · · 0 A2k





 .

The proof is presented in Lemma 2.26 of [13], the caseσ(i) =ifor some i, is also taken care in the proof. Now we give a sufficient condition for Q(Γ) to beD_θ⁰(D_θ(C^∗(Γ))), whereθ⁰ is an order 2 CQG automorphism of D_θ(C^∗(Γ)).

Proposition 2.21. Let Γbe a group withkgenerators{a₁, a₂,· · ·a_k} and define γ2l−1 := a_l, γ_2l := a⁻¹_l ∀ l = 1,2,· · ·, k. Now σ₁, σ₂, σ₃ are three distinct automorphisms of order 2 on the set {1,2,· · ·,2k−1,2k} and θ₁, θ₂, θ₃ are automorphisms of the group given by θ_j(γ_i) = γ_σ_j_(i) for all j = 1,2,3 and i= 1,2,· · ·,2k. We assume the following:

(1) B_i^(s):=U_i,σ_s_(i)6= 0∀ i, ands= 1,2,3alsoUi,j= 0∀j6∈ {σ_s(i), i}, (2) A_iB_j^(s) = B^(s)_j A_i = 0 ∀ i, j, s such that σ_t(i) 6= i, σ_t(j) 6= j ∀ t

where A_i =U_i,i,

(3) B_i^(s)B_j^(k) = B_j^(k)B_i^(s) = 0 ∀ i, j, s, k with s 6= k and σt(i) 6= i, σt(j)6=j ∀ t,

(4) All U_i,jU_i,j^∗ are central projections,

(5) There are well defined C^∗-isomorphisms π1, π₂^(s) from C^∗(Γ) to C^∗{A_i, i = 1,2,· · ·,2k} and C^∗{B_i^(s), i= 1,2,· · ·,2k} respectively where s= 1,2,3 such that

π1(λai) =Ai, π₂^(s)(λai) =B_i^(s) ∀ i.

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Furthermore, assume that using the group automorphisms we have two CQG automorphisms θ and θ⁰ of order 2 from C^∗(Γ) and D_θ(C^∗(Γ)) re- spectively defined by

θ(λx) =λ_θ₁_(x),

θ⁰(λ_x, λ_θ₁_(y)) = (λ_θ₂_(x), λ_θ₃_(y)) ∀ x, y∈Γ.

Then Q(Γ) will be D_θ⁰(D_θ(C^∗(Γ))) corresponding to the given automor- phisms. Moreover, the fundamental unitary takes the following form







A₁ B₁⁽¹⁾ 0 0 · · · B₁⁽²⁾ B₁⁽³⁾ B₂⁽¹⁾ A2 0 0 · · · B₂⁽³⁾ B₂⁽²⁾ 0 0 A₃ B₃⁽¹⁾ · · · 0 0 0 0 B₄⁽¹⁾ A4 · · · 0 0

... ...

B_2k−1⁽²⁾ B_2k−1⁽³⁾ 0 0 · · · A_2k−1 B⁽¹⁾_2k−1 B_2k⁽³⁾ B_2k⁽²⁾ 0 0 · · · B_2k⁽¹⁾ A_2k





 .

The proof is very similar to the Proposition 2.20, thus omitted. We end the discussion of Section 2 with the following easy observation which will be useful later.

Proposition 2.22. If U V = 0 for two normal elements in a C^∗-algebra then

U^∗V =V U^∗= 0, V^∗U =U V^∗=V U = 0.

Its proof is straightforward, hence omitted.

3. QISO computation of the braid group

In this section we will compute the quantum isometry group of the braid group with 3 generators. The group has a presentation

Γ =<a, b, c|ac=ca, aba=bab, cbc=bcb>.

Here S ={a, b, c, a⁻¹, b⁻¹, c⁻¹}.

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Theorem 3.1. Let Γ be the braid group with above presentation. Then Q(Γ, S)∼=D_θ⁰(D_θ(C^∗(Γ)))with the choices of automorphisms as in Propo- sition 2.21 given by:

θ1(a) =a⁻¹, θ1(b) =b⁻¹, θ1(c) =c⁻¹, θ2(a) =c, θ2(b) =b, θ2(c) =a, θ₃(a) =c⁻¹, θ₃(b) =b⁻¹, θ₃(c) =a⁻¹. Proof. Let the action α ofQ(Γ, S) be given by

α(λ_a) =λ_a⊗A+λ_a⁻¹⊗B+λ_b⊗C+λ_b⁻¹⊗D+λ_c⊗E+λ_c⁻¹⊗F, α(λ_a⁻¹) =λ_a⊗B^∗+λ_a⁻¹⊗A^∗+λ_b⊗D^∗+λ_b⁻¹⊗C^∗+λ_c⊗F^∗+λ_c⁻¹⊗E^∗,

α(λb) =λa⊗G+λ_a⁻¹⊗H+λb⊗I+λ_b⁻¹⊗J+λc⊗K+λ_c⁻¹⊗L, α(λ_b⁻¹) =λa⊗H^∗+λ_a⁻¹⊗G^∗+λb⊗J^∗+λ_b⁻¹⊗I^∗+λc⊗L^∗+λ_c⁻¹⊗K^∗,

α(λc) =λa⊗M+λ_a⁻¹⊗N+λ_b⊗O+λ_b⁻¹⊗P+λc⊗Q+λ_c⁻¹⊗R, α(λ_c⁻¹) =λ_a⊗N^∗+λ_a⁻¹⊗M^∗+λ_b⊗P^∗+λ_b⁻¹⊗O^∗+λ_c⊗R^∗+λ_c⁻¹⊗Q^∗. Then, the fundamental unitary is of the form







A B C D E F

B^∗ A^∗ D^∗ C^∗ F^∗ E^∗

G H I J K L

H^∗ G^∗ J^∗ I^∗ L^∗ K^∗

M N O P Q R

N^∗ M^∗ P^∗ O^∗ R^∗ Q^∗





 .

We need a few lemmas to prove the theorem.

Lemma 3.2. All the entries of the above matrix are normal.

Proof. First, using the condition α(λ_ac) = α(λ_ca) comparing the coefficients of λ_a2, λ_a⁻², λ_b2, λ_b⁻², λ_c2, λ_c⁻² on both sides we have

AM =M A, BN =N B, CO=OC, DP =P D, EQ=QE, F R=RF.

(3.1) Applying the antipode we get

AE=EA, BF =F B, GK=KG, HL=LH, M Q=QM, N R=RN.

(3.2)

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Similarly, from the relation α(λ_ac⁻¹) =α(λ_c⁻¹_a) following the same argument as above, one can deduce the following

AF =F A, BE=EB, GL=LG, HK=KH, N Q=QN, M R=RM.

(3.3) We observe AE^∗+F B^∗ = 0 by comparing the coefficient ofλ_ac⁻¹ in the expression of α(λ_a)α(λ_a⁻¹). This shows that AE^∗A^∗ = 0 as B^∗A^∗ = 0.

Thus, (AE)(AE)^∗ =AEE^∗A^∗ =E(AE^∗A^∗) = 0. Similarly, all the terms of the equations (3.1), (3.2) and (3.3) are zero.

Further, using the condition α(λ_a)α(λ_a⁻¹) = α(λ_a⁻¹)α(λ_a) = λ_e⊗1_Q one can deduce

AC^∗ =AD^∗ =CA^∗ =C^∗A=DA^∗ =D^∗A= 0, A^∗C=A^∗D=BD^∗=D^∗B =BC^∗ =B^∗C =C^∗B = 0.

Applying the antipode we have

AG^∗=G^∗A=AH=HA=BG=BH^∗=H^∗B =GB= 0.

Similarly from α(λb)α(λ_b⁻¹) =α(λ_b⁻¹)α(λb) =λe⊗1_Q one obtains CJ =J C =CI^∗ =I^∗C=C^∗I =IC^∗ =J^∗C^∗=C^∗J^∗ = 0,

DI =ID=DJ^∗ =J^∗D= 0.

Again using α(λc)α(λ_c⁻¹) =α(λ_c⁻¹)α(λc) =λe⊗1_Q we have, EL=LE=EK^∗=K^∗E= 0,

F K =KF =F L^∗ =L^∗F = 0.

Moreover, using the relation α(λ_aba) = α(λ_bab) we obtain α(λ_ab) = α(λ_baba⁻¹). From α(λ_ab) = α(λ_baba⁻¹) comparing the coefficients of λ_b2

and λ_b⁻² on both sides we obtain CI = DJ = 0. Now applying the antipode we get I^∗G^∗ = J H = 0. This implies GI =J H = 0. Again from α(λ_ab⁻¹) =α(λ_b⁻¹_a⁻¹_ba) and applying previous arguments we can deduce CJ^∗ =DI^∗ = 0. Applying antipode we getGJ =IH = 0. Now from the unitarity condition we know GG^∗+HH^∗+II^∗+J J^∗+KK^∗+LL^∗= 1.

This shows that G²G^∗ = G as we have already got GH = GI = GJ = GK = GL = 0. In a similar way, it follows that G^∗G² = G. Thus we can conclude that G is normal. Using the same argument as before we can show that H, I, J, K, Lare normal, i.e. all the elements of 3rd row are normal. Using the antipode the normality of C, D, O, P follows.

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Now we are going to show that A, B, E, F, M, N, Q, Rare normal too.

UsingAA^∗+BB^∗+CC^∗+DD^∗+EE^∗+F F^∗ = 1 we can write A=A(AA^∗+BB^∗+CC^∗+DD^∗+EE^∗+F F^∗)

=A²A^∗+ACC^∗+ADD^∗ (asAB=AE =AF = 0)

=A²A^∗+ (AC^∗)C+ (AD^∗)D (as C, Dare normal)

=A²A^∗ (as AC^∗=AD^∗ = 0).

Similarly A^∗A² =A, henceA is normal. Following exactly a similar line of arguments one can show the normality of the remaining elements.

Lemma 3.3. C=D=G=H =K =L=O=P = 0.

Proof. From the relation α(λac) = α(λca) equating the coefficients of λ_ba, λ_ab, λ_ab⁻¹, λ_b⁻¹_a on both sides we get AO = M C, CM = OA, AP = M D, CN = OB. This implies that CM M^∗ = OAM^∗ = 0, CN N^∗ = OBN^∗ = 0 as AM^∗ = BN^∗ = 0. Similarly one can obtain CQQ^∗ = CRR^∗ = 0. Now using (AA^∗+BB^∗+GG^∗+HH^∗+M M^∗+N N^∗) = 1 we have

C=C(AA^∗+BB^∗+GG^∗+HH^∗+M M^∗+N N^∗)

=C(AA^∗+BB^∗+GG^∗+HH^∗) (asCM M^∗ =CN N^∗ = 0)

=C(GG^∗+HH^∗) (as CAA^∗ =CA^∗A= 0, CBB^∗=CB^∗B = 0).

Moreover, we have

C=C(EE^∗+F F^∗+KK^∗+LL^∗+QQ^∗+RR^∗)

=C(KK^∗+LL^∗+QQ^∗+RR^∗) (asCE^∗ =CF^∗ = 0)

=C(KK^∗+LL^∗) (asCQQ^∗ =CRR^∗ = 0).

Using the above equations we get that C(KK^∗+LL^∗)(GG^∗+HH^∗) = C(GG^∗ +HH^∗) = C = 0 (as KG = KH = LG = LH = 0). Simi- larly, we can find D = 0. Then we have G = H = 0 by using the antipode. Moreover, AO = M C = 0, AP = OB = BO = 0. This gives us O = (A^∗A+B^∗B+M^∗M +N^∗N)O = 0. Similarly, we get P = 0, K =

L= 0.